It sounds like you need to work on your grip. I have big hands too, it turned out to be a problem with how I was holding the neck of the instrument. You should see about having a tutor help you. It should not be causing you any pain. Wrist pain is almost always due to a grip that is too strong, keep that up and you'll get tendonitis.
But the thing about the violin (and stringed instruments in general) is that technique matters more than anything else. It's not like keyboards or woodwinds (no hate on either of those, of course, being a piano, pipe organ, and clarinet player as well as a violinist) where it's just pressing a button, exactly how you put your finger on the strings will matter a lot and there is more than one valid way to approach this. A few suggestions:
1.) Try adjusting your chin and shoulder rests. When I started violin I was having wrist and grip problems everywhere. I tried everything and spent weeks trying to figure it out, eventually I broke down and had a meeting with a stringed instrument professor at my university, he took one look at my form and told me my shoulder rest was too low, which caused me to not hold the violin properly with my neck, and I was compensating with my left hand by keeping it at an angle that made it impossible to use the tips of my fingers to press the strings down. Spend some time adjusting your shoulder rest until you get a position that has the strings PERFECTLY parallel to the floor while keeping your left arm completely relaxed. I had been playing for over a year before I caught this, and I had the same problems with bumping strings with my fingers. There should be no pain or discomfort whatsoever, it should actually feel quite pleasant when you're holding it properly...weightless, almost.
2.) Vibrato is hard to learn, it just takes time to build up the muscular ability for it. Don't worry if you're not getting it right away.
